A piece of string is 40 cm long. It is cut into three pieces. The longest piece is three times as long as the middle-sized piece, and the shortest piece is 23 cm shorter than the longest piece. Find the length of the shortest piece (in cm).
A house contains a diamond, gold, and silver. The ratio of all excluding diamond to all excluding gold and all excluding silver is 47:27:17. Find the part of gold in it.
Abhijit and Saurav can do work in 15 days, Piyush and Saurav in 20, and Abhijit and Piyush in 10 days. In how many days Piyush will complete the work alone?
How many kilograms of wheat costing Rs. 8/kg must be mixed with 36 kg of rice costing Rs. 5.40/kg so that 20% gain may be obtained by selling the mixture at Rs. 7.20/kg?
If the numerator of a fraction is increased by 400% and the denominator is increased by 500%, the resultant fraction is \( \frac{20}{27} \). What is the original fraction?